Heat 1 tablespoon butter in a large skillet over medium heat until melted. Add bread and swirl to coat evenly in butter. Cook, shaking pan and moving bread occasionally until light golden brown on one side. Transfer bread to cutting board, browned-side up.
Place 1 slice of cheese on top of browned surface of each slice of bread. Top with thanksgiving leftovers, layering as desired. Close sandwiches so that non-browned side is facing outwards.
Heat half tablespoon butter over medium heat until melted and add sandwich. Cook, moving occasionally, until deep golden brown, about 3 minutes. Remove sandwich and melt remaining half tablespoon butter. Return sandwich to skillet to brown second side, about 3 minutes longer. Serve with hot gravy.